In practice we do not observe the nonlinear matter density field, but instead its tracers.
Modeling the connection between tracers and the underlying density field is a complex task, and there are several prevailing approaches to this problem.
The large-scale bias approach extends the philosophy of perturbation theory to parameterize the tracer field as a linear combination of locally-leading gravitational observables [1]}.
These models have been successful on large scales, but face the same issue of PT for modeling dark matter, namely the presence of a nonlinear scale that characterizes the dark matter dynamics [2]}.
On top of this fundamental limit, there is an additional limiting scale, the nonlocality scale, which characterizes the formation of the tracer (e.g. halos/galaxies) and is not necessarily coincident with the nonlinear scale.
Recently, some pragmatic semi-analytic models have partially circumvented this issue by considering bias with respect to a fitting function for the non-linear matter power [3]}, using bias templates measured from simulations [4]}, or by extending the halo model via functions fit to simulations [5]}.
In the halo occupation model approach, luminous tracers are assigned to a catalog of dark matter halos via a prescription for stochastically populating the halos - this is the Halo Occupation Distribution (HOD) framework [6]}, [7]}, [8]}.
HOD models, usually combined with halos found in N-body simulations, are used for some modern analyses that include small-scale galaxy clustering [9]}, [10]}.
There are also more involved models of the galaxy-halo connection that are less frequently used in cosmological analyses [11]}.
Finally, hydrodynamic simulations that include baryonic/gas physics attempt to model galaxy formation more directly, albeit with stochastic subgrid models mixed in [12]}.
These simulations are extremely computationally expensive to run, and typically cannot be run at the sufficient number of realizations or volume to be relevant for modern redshift surveys.
